Dynamic Programming

Dynamic Programming is a technique used to solve problems by breaking them down into smaller overlapping subproblems and solving each subproblem only once. It is particularly useful for problems that exhibit the optimal substructure property, which means that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.

Dynamic Programming can improve the efficiency of recursive algorithms by eliminating duplicate work through memoization or by solving the subproblems in a bottom-up manner and building the solution incrementally.

Fibonacci Sequence Example

Let's take a look at an example to understand how dynamic programming works. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The sequence starts with 0 and 1.

Here's a Java code snippet that computes and prints the first 10 numbers in the Fibonacci sequence using dynamic programming:

1{code}

The code starts by defining an array dp to store the Fibonacci numbers. It initializes the base cases dp[0] = 0 and dp[1] = 1. Then, it uses a loop to compute the remaining Fibonacci numbers by summing up the two preceding ones. Finally, it prints the computed Fibonacci numbers.

By using dynamic programming, we avoid redundant calculations and improve the efficiency of computing the Fibonacci sequence.

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class Main {

  public static void main(String[] args) {

    // replace with your Java logic here

    int n = 10;

    int[] dp = new int[n + 1];

​

    // base cases

    dp[0] = 0;

    dp[1] = 1;

​

    // compute fibonacci numbers using dynamic programming

    for (int i = 2; i <= n; i++) {

      dp[i] = dp[i - 1] + dp[i - 2];

    }

​

    // print fibonacci numbers

    for (int i = 0; i <= n; i++) {

      System.out.print(dp[i] + " ");

    }

  }

}